Full Text
UDC 517.944+517.63
MATHEMATICS
O. A. DYSHIN
AN OPERATIONAL METHOD FOR SOLVING MIXED PROBLEMS FOR LINEAR DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE WITH DISCONTINUOUS COEFFICIENTS
(Presented by Academician A. A. Dorodnitsyn, March 30, 1967)
The mixed problem (1)—(3) was investigated in the general case of complex-valued functions \(c_2^{(i)}(x)\), \(-\pi/2 < \arg c_2^{(i)}(x) < \pi/2\) \((i=1,\ldots,n)\), by the contour-integral method in the monograph of M. L. Rasulov \((^1)\). We consider an operational scheme of solution and give a justification of the operational method for solving problem (1)—(3) on the half-line \(t>0\). The realization of the obtained operational formulas of solution (8) is connected with the inversion of Laplace integrals. Numerical methods for inverting these integrals have been sufficiently developed \((^5,\) Ch. 5).
- Consider the problem of finding solutions of the equations
\[ \frac{\partial u^{(i)}(x,t)}{\partial t} = \sum_{l=0}^{2} c_l^{(i)}(x)\frac{\partial^l u^{(i)}(x,t)}{\partial x^l} + f^{(i)}(x,t), \tag{1} \]
\[ x\in(a_i,b_i),\quad t\in(0,T],\quad (i=1,\ldots,n) \]
under the boundary conditions
\[ \sum_{i=1}^{n}\sum_{l=0}^{1} \left\{ \alpha_{sl}^{(i)}\!\left(\frac{\partial}{\partial t}\right) \left. \frac{\partial^l u^{(i)}(x,t)}{\partial x^l} \right|_{x=a_i} + \beta_{sl}^{(i)}\!\left(\frac{\partial}{\partial t}\right) \left. \frac{\partial^l u^{(i)}(x,t)}{\partial x^l} \right|_{x=b_i} \right\} = \gamma_s, \]
\[ t\in(0,T],\quad (s=1,2,\ldots,2n) \tag{2} \]
and the initial conditions
\[ u^{(i)}(x,+0)=0,\quad x\in(a_i,b_i)\quad (i=1,\ldots,n), \tag{3} \]
where
\[ \alpha_{sl}^{(i)}\!\left(\frac{\partial}{\partial t}\right) = \alpha_{sl0}^{(i)}+\alpha_{sl1}^{(i)}\frac{\partial}{\partial t}, \qquad \beta_{sl}^{(i)}\!\left(\frac{\partial}{\partial t}\right) = \beta_{sl0}^{(i)}+\beta_{sl1}^{(i)}\frac{\partial}{\partial t}, \]
\(\alpha_{slk}^{(i)}\), \(\beta_{slk}^{(i)}\) \((l,k=0,1)\), \(\gamma_s\) \((s=1,2,\ldots,2n)\) are, generally speaking, complex constants; \((a_i,b_i)\) are mutually nonintersecting intervals having common boundary points and contained in the basic interval \((a_1,b_n)\) \((n\ge 1)\).
Problem (1)—(3) is reduced to two problems: problem A—the determination of solutions \(v^{(i)}(x,t)\) of the homogeneous equations corresponding to equation (1), under the boundary conditions (2) and the initial conditions (3), and problem B—the determination of solutions \(w^{(i)}(x,t)\) of equations (1) satisfying the homogeneous boundary conditions obtained from (2) when \(\gamma_s=0\) \((s=1,2,\ldots,2n)\), and the initial conditions (3).
By a solution of problems A and B we shall mean functions \(v^{(i)}(x,t)\) and \(w^{(i)}(x,t)\) \((i=1,\ldots,n)\) that are continuous for \(x\in(a_i,b_i)\), \(t\in[0,T]\), and for \(x\in[a_i,b_i]\), \(t\in(0,T]\) \((0<T<\infty)\), all derivatives entering into (1) are continuous with respect to \(x\in(a_i,b_i)\) and \(t\in(0,T]\), while the derivatives entering into (2) are continuous with respect to \(x\in[a_i,b_i]\) and \(t\in(0,T]\).
We shall assume the following conditions to be fulfilled:
a) the functions \(c_l^{(i)}(x)\) \((l=0,1,2)\) are \(6-l\) times continuously differentiable on \([a_i,b_i]\), with \(c_0^{(i)}(x)\), \(c_1^{(i)}(x)\) complex-valued, and \(c_2^{(i)}(x)\) real-valued functions, and \(c_2^{(i)}(x)>0\) for all \(x\in[a_i,b_i]\);
b) for any \(t \in [0,T]\) the functions \(\partial^k f^{(i)}(x,t)/\partial t^k\) \((k=0,1,2)\) and \(\partial f^{(i)}(x,t)/\partial x\) are continuously differentiable with respect to \(x\) on \([a_i,b_i]\); \(f^{(i)}(x,t)\) are complex-valued functions.
Taking for \(z=\sqrt{\lambda}\) that branch for which \(z \in \widetilde{\Pi}_1\{z:0 \leq \arg z \leq \pi/2\}\) when \(\lambda \in \Pi_1\{\lambda:0 \leq \arg\lambda \leq \pi\}\) and \(z \in \widetilde{\Pi}_2\{z:-\pi/2 \leq \arg z \leq 0\}\) when \(\lambda \in \widetilde{\Pi}_2\{\lambda:-\pi \leq \arg\lambda \leq 0\}\), put
\[ A_{sk}^{(i)}(z)=\sum_{l=0}^{1}\alpha_{sl}^{(i)}(z^2)\bigl(z\varphi_k^{(i)}(a_i)\bigr)^l,\quad B_{sk}^{(i)}(z)= \]
\[ =\sum_{l=0}^{1}\beta_{sl}^{(i)}(z^2)\bigl(z\varphi_k^{(i)}(b_i)\bigr)^l,\quad \varphi_k^{(i)}(x)=(-1)^k\bigl(c_2^{(i)}(x)\bigr)^{-1/2} \quad (k=1,2;\ s=1,2,\ldots,2n). \]
Denoting by \(A_k^{(i)}(z), B_k^{(i)}(z)\) \((k=1,2)\) the columns with elements \(A_{1k}^{(i)}(z),\ldots,A_{2n,k}^{(i)}(z)\) and \(B_{1k}^{(i)}(z),\ldots,B_{2n,k}^{(i)}(z)\), respectively, we formulate conditions A and B\(_{(k)}\) \((k=0,1,2)\).
A. The determinants
\[ \left|A_1^{(1)}(z)\ B_2^{(1)}(z)\ A_1^{(2)}(z)\ B_2^{(2)}(z)\ \ldots\ A_1^{(n)}(z)\ B_2^{(n)}(z)\right|, \]
\[ \left|B_1^{(1)}(z)\ A_2^{(1)}(z)\ B_1^{(2)}(z)\ A_2^{(2)}(z)\ \ldots\ B_1^{(n)}(z)\ A_2^{(n)}(z)\right| \]
are polynomials in \(z\) of the same degree \(d \geq 0\), different from the identically zero polynomial. All determinants of order \(2n\) formed from other combinations of the columns \(A_k^{(i)}(z), B_k^{(i)}(z)\) \((k=1,2;\ i=1,\ldots,n)\) are polynomials in \(z\) of degree not exceeding \(d\).
B\(_{(k)}\). Suppose that in condition A \(d \geq k\) and all determinants of order \(2n-1\) formed from the elements of the matrix
\[ \bigl(A_1^{(1)}(z)\ A_2^{(1)}(z)\ \ldots\ A_1^{(n)}(z)\ A_2^{(n)}(z)\ B_1^{(1)}(z)\ B_2^{(1)}(z)\ \ldots\ B_1^{(n)}(z)\ B_2^{(n)}(z)\bigr) \]
are polynomials in \(z\) of degree not exceeding \(d-k\).
To the original problem (1)—(3) we assign the boundary-value problem with a complex parameter (the spectral problem)
\[ \sum_{l=0}^{2} c_l^{(i)}(x)\frac{d^l y^{(i)}}{dx^l}-\lambda y^{(i)}=0,\quad a_i<x<b_i\quad (i=1,\ldots,n); \tag{4} \]
\[ \sum_{i=1}^{n}\sum_{l=0}^{1} \left\{ \left.\alpha_{sl}^{(i)}(\lambda)\frac{d^l y^{(i)}}{dx^l}\right|_{x=a_i} + \left.\beta_{sl}^{(i)}(\lambda)\frac{d^l y^{(i)}}{dx^l}\right|_{x=b_i} \right\} =\gamma_s \quad (s=1,2,\ldots,2n). \tag{5} \]
Let \(y_k^{(i)}(x,\lambda)\) \((k=1,2)\) be a fundamental system of particular solutions of the \(i\)-th equation (4), consisting of functions which, together with their first derivatives with respect to \(x\), are entire functions of the parameter \(\lambda\) (see Poincaré’s theorem \((^4)\), p. 28). Denote
\[ u_{sk}^{(i)}(\lambda)= \sum_{l=0}^{1} \left\{ \left.\alpha_{sl}^{(i)}(\lambda)\frac{d^l y_k^{(i)}}{dx^l}\right|_{x=a_i} + \left.\beta_{sl}^{(i)}(\lambda)\frac{d^l y_k^{(i)}}{dx^l}\right|_{x=b_i} \right\} \]
\[ (k=1,2;\ i=1,\ldots,n;\ s=1,2,\ldots,2n). \]
Now making use of the auxiliary notation from \((^1)\), we introduce for consideration, by formulas (6.2.5), (6.2.12)—(6.2.16) of \((^1)\), the functions \(w^{(i)}(x,\lambda)\), \(\Delta^{(i)}(x,\lambda)\), \(\Delta^{(i,j)}(x,\xi,\lambda)\), \(g^{(i,j)}(x,\xi,\lambda)\), \(\Delta(\lambda)\), \(G^{(i,j)}(x,\xi,\lambda)\). Following Birkhoff \((^3)\), we shall denote a sum of the form \(f(x)+E(x,z)/z\), where \(E(x,z)\) is continuous in \(x\) and bounded for \(|z|\geq R\) (\(R\) is a sufficiently large positive number), by the symbol \([f(x)]\). Then, applying Tamarkin’s theorem on the asymptotic representation of a certain fundamental system of particular solutions to each \(i\)-th equation of (4) (see \((^2)\), theorem 3), for the determinant \(\widetilde{\Delta}(z)=\Delta(z^2)\) we find, for all sufficiently large in modulus \(z \in \widetilde{\Pi}\), \(\widetilde{\Pi}=\widetilde{\Pi}_1+\widetilde{\Pi}_2\), the asymptotic formula
\[ \widetilde{\Delta}(z)=z^{d-n}H(z)\prod_{i=1}^{n}\psi^{(i)}(a_i z^2), \]
\[ H(z)=\sum_{\mu=1}^{\sigma}[M_\mu]\exp(m_\mu z),\qquad m_1<\cdots<m_\sigma, \]
\[ -m_1=m_\sigma=\sum_{i=1}^{n}\int_{a_i}^{b_i}\bigl(c_2^{(i)}(x)\bigr)^{-1/2}\,dx,\qquad M_1\ne0,\qquad M_\sigma\ne0. \]
By Lemma 1 of \(({}^{1})\), all zeros of the function \(H(z)\) are located in the \(z\)-plane in a certain strip \(D_h\), bounded by the straight lines \(\operatorname{Re}(z)=\pm h\) \((h>0)\).
With the aid of the concrete asymptotics of the solution of the spectral problem (4)—(5) and of the existence and well-posedness theorems for the solutions of problems A and B (Theorems 24, 25, 28 of \(({}^{1})\)), we prove the following theorems.
Theorem 1. Under conditions a), b) and A, problem B has a unique solution \(w^{(i)}(x,t)\), depending continuously on the right-hand sides of equations (1) and representable for all \(x\in[a_i,b_i]\), \(t\in(0,T]\) and \(x\in(a_i,b_i)\), \(t\in(0,T]\) by the formula
\[ w^{(i)}(x,t)= -\frac{1}{2\pi\sqrt{-1}}\lim_{\omega\to\infty} \int_{\eta-\omega\sqrt{-1}}^{\eta+\omega\sqrt{-1}} e^{\lambda t}\,d\lambda \sum_{j=1}^{n}\int_{a_j}^{b_j}G^{(i,j)}(x,\xi,\lambda)\times \]
\[ \times\int_{0}^{t}e^{-\lambda\tau}f^{(j)}(\xi,\tau)\bigl(c_2^{(j)}(\xi)\bigr)^{-1}\,d\tau\,d\xi, \qquad \eta>h^2\quad (i=1,\ldots,n). \tag{6} \]
All derivatives of \(w^{(i)}(x,t)\) entering into problem B are obtained by differentiating under the sign of the contour integral in (6).
Let us call \(A_{(0)}\), \(A_{(1)}\), \(A_{(2)}\) the problems A with boundary conditions (2), respectively: 1) containing no derivatives with respect to \(t\); 2) containing the derivative \(\partial u^{(i)}/\partial t\) for at least one value of \(i\) and \(s\); 3) containing the derivative \(\partial^2 u^{(i)}/\partial t\,\partial x\) for at least one value of \(i\) and \(s\).
Theorem 2. Problem \(A_{(k)}\) has, under conditions a), A, \(B_{(k)}\), a unique solution, depending continuously on the right-hand sides of the boundary conditions (2) and representable for all \(x\in[a_i,b_i]\), \(t>0\) and \(x\in(a_i,b_i)\), \(t\ge0\) by the formula
\[ v^{(i)}(x,t)= \frac{1}{2\pi\sqrt{-1}}\lim_{\omega\to\infty} \int_{\eta-\sqrt{-1}\omega}^{\eta+\sqrt{-1}\omega} e^{\lambda t}\frac{\Delta^{(i)}(x,\lambda)}{\lambda\Delta(\lambda)}\,d\lambda, \qquad \eta>h^2\quad (i=1,\ldots,n) \]
\[ \tag{7} \]
All derivatives of the solution \(v^{(i)}(x,t)\) entering into problem \(A_{(k)}\) are obtained by differentiating under the sign of the contour integral in (7).
- Usually, when solving problem (1)—(3) on the half-line \(t>0\) by the operational method (with the aid of the one-sided Laplace transform with respect to the variable \(t\)), one assumes that the right-hand sides of equations (1) are Laplace-transformable with respect to \(t\) and makes the following assumptions: 1) the solution of the problem and all derivatives of the solution entering into the formulation of the problem are Laplace-transformable; 2) the operations of differentiation \(\partial/\partial x\), \(\partial^2/\partial x^2\) commute with \(L_t\) (\(L_t\) is the Laplace integral); 3) the operations of passage to the limit as \(x\to a_i\pm0\), \(x\to b_i\pm0\) commute with \(L_t\). Naturally, by this route only a formal solution can be obtained. Thus, for example, problem (1)—(3) is solved in \(({}^{6})\) with coefficients \(c_l^{(i)}(x)\) \((l=0,1,2;\ n=2)\), independent of \(x\), and with boundary conditions containing no derivatives with respect to \(t\).
Let \(S\{a,b\}\) be the set of all functions \(\Phi(x,t)\), defined for \(x\in[a,b]\) (or \(x\in(a,b)\)), \(0\le t<\infty\), integrable with respect to \(t\) on every finite interval \([0,T]\) for any \(x\in[a,b]\) (respectively, \(x\in(a,b)\)), and for which the Laplace integral \(L_t\) converges at some point \(\lambda_0=\lambda_0(\Phi)\) uniformly with respect to \(x\in[a,b]\) (respectively, \(x\in\)
\(\in (a,b))\). To each fixed function \(\Phi(x,t) \in S\{a,b\}\) there corresponds the set \(E\{\Phi,a,b\}\) of all real values \(\lambda\) for which \(L_t\) converges uniformly with respect to \(x\). The lower bound of the set \(E\{\Phi,a,b\}\) will be called the abscissa of convergence of the integral \(L_t\{\Phi(x,t)\}\).
Theorem 3. Under conditions a), A, \(\mathrm{B}_{(k)}\), the solution \(v^{(i)}(x,t)\) of problem \(\mathrm{A}_{(k)}\), together with all derivatives entering this problem, belongs to the set \(S\{a_i,b_i\}\). The abscissa of convergence of their Laplace integrals does not exceed \(h^2\).
Theorem 4. Let the right-hand sides \(f^{(i)}(x,t)\) of equations (1) be continuous for \(x \in [a_i,b_i]\), \(t>0\), and let, on the interval \([a_i,b_i]\), all the functions
\[ \frac{\partial^{s+m}}{\partial t^s \partial x^m} f^{(i)}(x,t) \]
\((m=0,1,2;\ s=0,1)\) belong to the set \(S\{a_i,b_i\}\) with abscissa of convergence \(\mu^{(i)} \geqslant 0\) of their Laplace integrals.
Then, under conditions a), A, the solution \(w^{(i)}(x,t)\) of problem B, together with all derivatives entering this problem, belongs to the set \(S\{a_i,b_i\}\). The abscissa of convergence of their Laplace integrals does not exceed \(\max(\mu^{(i)},h^2)\).
It is clear that, under the conditions of Theorems 3 and 4, assumptions 1), 2), 3) are certainly satisfied for problem (1)—(3), which makes it possible to solve this problem directly by the operational method.
As a result, for the solution of the problem,
\[ u^{(i)}(x,t) = \frac{1}{2\pi\sqrt{-1}} \lim_{\omega\to\infty} \int_{\eta-\omega\sqrt{-1}}^{\eta+\omega\sqrt{-1}} e^{\lambda t} \times \]
\[ \times \left( \frac{\Delta^{(i)}(x,\lambda)}{\lambda\Delta(\lambda)} - \sum_{j=1}^{n} \int_{a_j}^{b_j} G^{(i,j)}(x,\xi,\lambda)\, \bar f^{(j)}(\xi,\lambda)\, \bigl(c_2^{(j)}(\xi)\bigr)^{-1} d\xi \right)d\lambda, \tag{8} \]
\[ \eta>\max(\mu^{(i)},h^2) \qquad (i=1,\ldots,n), \]
which is valid for all \(x \in [a_i,b_i]\), \(t>0\), and \(x \in (a_i,b_i)\), \(t \geq 0\).
In conclusion I express my deep gratitude to my scientific adviser, Prof. V. A. Ditkin, for his constant attention to the work and valuable comments.
Computing Center
Academy of Sciences of the USSR
Received
23 III 1967
CITED LITERATURE
- M. L. Rasулов, The method of the contour integral and its application to the study of problems for differential equations, “Nauka,” 1964.
- Ya. D. Tamarkin, On some general problems of the theory of ordinary differential equations and the expansion of arbitrary functions in series, Petrograd, 1917.
- G. D. Birkhoff, Trans. Am. Math. Soc., 9, 210 (1908).
- É. Goursat, A Course of Mathematical Analysis, 3, part 1, Moscow—Leningrad, 1938.
- V. A. Ditkin, A. P. Prudnikov, Results of Science, Mathematical Analysis, 1964, Moscow, 1966, p. 7.
- A. V. Ivanov, in: Thermal Physics in Foundry Production, Minsk, 1963, p. 11.